## 5.3The Definite Integral

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### OBJECTIVES

When you finish this section, you should be able to:

1. Define a definite integral as the limit of Riemann sums (p. 11)
2. Find a definite integral using the limit of Riemann sums (p. 14)

The area $$A$$ under the graph of $$y = f(x)$$ from $$a$$ to $$b$$ is obtained by finding $$$\boxed{\bbox[#FAF8ED,5pt]{{A=\lim\limits_{n\rightarrow \infty}s_{n}=\lim\limits_{n\rightarrow \infty }{\sum\limits_{i=1}^{n}} {f}(c_{i}) \Delta x=\lim\limits_{n\rightarrow \infty }S_{n}=\lim\limits_{n\rightarrow \infty }{\sum\limits_{i=1}^{n}} {f}(C_{i}) \Delta x }}}$$$

### THEOREM The Integral of the Sum of Two Functions

If two functions $$f$$ and $$g$$ are continuous on the closed interval $$[a,b]$$, then $\begin{equation*} \boxed{\bbox[5pt]{\int_{a}^{b}[ f(x)+g(x)] \,dx=\int_{a}^{b}f(x)\,dx+\int_{a}^{b}g(x)\,dx}}\tag{1} \end{equation*}$